Lattice model for pressure problems in two-dimensional granular columns

Abstract

In order to make it easier to investigate some problems such as the mechanism of Janssen effect and the stress distribution in granular medium, we simplify a granular column into a lattice system, in which a lattice point represents a small lump of granular medium and only neighbor interactions are considered. To study the disordered granular columns, a force propagation lattice model determined by the absorption coefficient p and the lateral transfer coefficient q is proposed, and this model is analyzed from the theoretical view. Firstly, the equation of force propagation in the matrix form is given, and this equation is determined by a tridiagonal matrix A(p,q) that is called transfer coefficient matrix. Based on the force transfer equation, the bottom force distribution varying with the top force distribution and the layer of lattice system is deduced, and its analytical solution refers to the similarity diagonalization of matrix A(p, q). Then, a method based on the second order difference equations is proposed to obtain the eigenvalues and eigenvectors of the transfer coefficient matrix. The eigenvalues and eigenvectors of A(p, q) can be rigorously deduced for a typical case, and with these results the pressure distribution relationship between the top and the bottom of the container is given. Based on these theoretical expressions, the relationship between the effective mass and the total mass of granular medium is deduced, and it means that the force propagation model and the Janssen model can lead to similar results. Moreover, the bottom stress distribution is calculated without the top load. Calculations show that the stress distribution reaches a maximum at the center bottom and drops down to either side. Finally, numerical calculations are performed to investigate the effects of parameters p and q on the relation between bottom pressure and packing height. Numerical results show that the saturated value of pressure decreases while parameter p or q increases.